We study the composition of an arbitrary number of Fourier integral operators A j , j = 1, . . . , M, M ≥ 2, defined through symbols belonging to the so-called SG classes. We give conditions ensuring that the composition A 1 • · · · • A M of such operators still belongs to the same class. Through this, we are then able to show well-posedness in weighted Sobolev spaces for first order hyperbolic systems of partial differential equations in SG classes, by constructing the associated fundamental solutions. These results expand the existing theory for the study of the properties "at infinity" of the solutions to hyperbolic Cauchy problems on R n with polynomially bounded coefficients.

We consider the Cauchy problem for linear and quasilinear symmetrizable hyperbolic systems with coefficients depending on time and space, not smooth in t and growing at infinity with respect to x. We discuss well-posedness in weighted Sobolev spaces, showing that the non-Lipschitz regularity in t has an influence not only on the loss of derivatives of the solution but also on its behaviour for |x| → ∞. We provide examples to prove that the latter phenomenon cannot be avoided.

We consider p-evolution equations in (t,x) with real characteristics. We give sufficient conditions for the well-posedness of the Cauchy problem in Sobolev spaces, in terms of decay estimates of the coefficients as the space variable x goes to infinity

We study random-field solutions of a class of stochastic partial di↵erential equations, involving operators with polynomially bounded coe cients. We consider linear equations under suitable hyperbolicity hypotheses, and we provide conditions on the initial data and on the stochastic term, namely, on the associated spectral measure, so that these kind of solutions exist in suitably chosen functional classes. We also give a regularity result for the expected value of the solution.

In this article we show the existence of a random-field solution to linear stochastic partial differential equations whose partial differential operator is hyperbolic and has variable coefficients that may depend on the temporal and spatial argument. The main tools for this, pseudo-differential and Fourier integral operators, come from microlocal analysis. The equations that we treat are second-order and higher-order strictly hyperbolic, and second-order weakly hyperbolic with uniformly bounded coefficients in space. For the latter one we show that a stronger assumption on the correlation measure of the random noise might be needed. Moreover, we show that the well-known case of the stochastic wave equation can be embedded into the theory presented in this article.